Efforts to get Exxon Mobil Corp. CEO Rex Tillerson to step down as chairman of the board are growing heated as members of the prestigious Rockefeller family tell other large shareholders they’ve been routinely blocked from meeting with directors.

Exxon, meanwhile, is fighting back, sending letters to shareholders telling them to reject the effort.

Some shareholders say Exxon’s corporate secretary and vice president of investor relations has also been setting up meetings to talk with them about the proposal, but Exxon spokesman Alan Jeffers says the meetings are to discuss the company’s earnings.

Attempts to separate the CEO and chairman roles at Exxon got a boost last month when the normally private descendents of John D. Rockefeller – founder of Standard Oil, the company that grew into Exxon – joined the fight.

The Rockefellers want to talk with the board about exploring alternatives to fossil fuels amid worry that profits at Exxon could suffer if the demand for oil drops or if the US government adopts new environmental regulations that could curb oil consumption.

The Rockefeller family has been meeting with a series of large investors to win support for their effort, including mutual fund firms Barclays Global Investors, State Street and Putnam.

In meetings, they’ve raised questions about the board’s independence and about management’s perceived role as “gatekeepers to the board,” said one meeting attendee.

Peter O’Neill, head of the Rockefeller committee on Exxon, said no one has accused Exxon’s management team of controlling the board.

“Since we can’t really talk to the board, we can’t really know,” he said.

Still, that’s the view some shareholders are walking away with.

“The culture is very insular, it’s very closed, and there’s an inability to access the board,” said an official with one of the large shareholders who talked to the Rockefellers. “They want greater leadership from the board.”

Tillerson, who took the reins of Exxon in 2006, was widely expected to be an improvement on his notoriously brash predecessor, Lee Raymond, who’s known for once making an attendee of the company’s annual meeting cry – and for his $351 million retirement package.

And while Tillerson is often described as nicer to investors than Raymond was, some shareholders say he’s continued the tradition of keeping the board under lock and key.

“That’s just the way this company is run. Management has very centralized authority,” said Robert A. G. Monks, the original filer of the proposal to separate the roles at Exxon.